I have been learning about Lie groups and there is a question that has been in the back of my mind for a while. I will try to formulate it with an analogy. On a differential manifold with a metric structure, the metric structure induces an isomorphism between vectors and 1-forms (the musical isomorphisms). Yet, we have a natural structure (here I use natural to indicate that its definition does not necessitate any additional assumption other than the manifold being a differentiable one), the Lie derivative, that can distinguish between vectors and 1-forms. In other words, for a general metric, the Lie derivative and the musical isomorphism do not commute. Is there something similar for Lie groups? For example, SO(2) and U(1) are isomorphic. So are they really distinct groups? In other words, is there a way to distinguish them or are they just representations of the same abstract group?
I guess my question is what exactly uniquely defines a group? What natural (as defined above) structures can be used to differentiate abstract groups?